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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 125, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2006/125\hfil Optimal control of an epidemic]
{Optimal control of an epidemic through educational campaigns}
\author[C. Castilho\hfil EJDE-2006/125\hfilneg]
{C\'esar Castilho} % in alphabetical order
\address{Departamento de Matem\'atica \\
Universidade Federal de Pernambuco\\
Recife, PE CEP 50740-540 Brazil \newline
and The Abdus Salam ICTP \\
Strada Costiera 11 Trieste 34100 Italy}
\email{castilho@dmat.ufpe.br}
\date{}
\thanks{Submitted September 21, 2005. Published October 11, 2006.}
\subjclass[2000]{92D30, 93C15, 34H05}
\keywords{Epidemic; optimal control; educational campaign}
\begin{abstract}
In this work we study the best strategy for educational campaigns
during the outbreak of an epidemic. Assuming that the epidemic is
described by the simplified SIR model and that the total time of
the campaign is limited due to budget, we consider two possible
scenarios. In the first scenario we have a campaign oriented to
decrease the infection rate by stimulating susceptibles to have a
protective behavior. In the second scenario we have a campaign
oriented to increase the removal rate by stimulating the infected
to remove themselves from the infected class. The optimality is
taken to be to minimize the total number of infected by the end of
the epidemic outbreak. The technical tool used to determine the
optimal strategy is the Pontryagin Maximum Principle.
\end{abstract}
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\section{Introduction}
In this work we study the best strategy for educational campaigns
during the outbreak of an epidemic. We assume that the epidemic is
described by the simplified SIR model \cite{thi} and also assume
that the total time of the campaign is budget limited. Optimality is
measured minimizing the total number of infected at the end of the
optimal outbreak. If we cannot make a campaign during all the
epidemic time, what is the optimal way of using the time we have?
How many campaigns should we make? What should be their intensities?
When should they start? The difficult point is, of course, how to
model the effect of the campaign on the spread of the epidemic. Here
we face two problems: first, the model must be intuitively plausible
and second, it must be mathematically tractable.
With respect to the first requirement
we will model the campaign effects by reducing the rate at which
the disease is contracted from an average individual \emph{during}
the campaign (called shortly infection rate). We justify this with
an example: suppose during a flu outbreak one starts a campaign
orienting susceptibles to avoid contracting the virus (assuming
some protective behavior, e.g., washing hands, avoiding close
environments, etc.). The effect of the campaign will be that the
probability of a susceptible contracting the virus will decrease.
The same reasoning applied to a campaign oriented to
the infected (e.g. stimulating quarantine) will be modelled
increasing the rate at which an average individual leaves the
infective rate (called shortly removal rate). With respect to the
second requirement we assume, for mathematical simplicity, that
this reduction (increase) is bounded below (above) and the
campaigns cost are linear on the controls. With those hypotheses
the problem renders itself to analytical treatment and we can
prove the main facts about the optimal campaign. The theorems of
section \ref{C2} reduce the dimension of the optimal problem
allowing a complete numerical study of the problem.
Application of control theory to epidemics is a very large field.
A comprehensive survey of control theory applied to epidemiology
was performed by Wickwire \cite{wick1}. Many different models with
different objective functions have been proposed (see
\cite{gupta,hethe,morton} and more recently
\cite{horst,zaric}). A major difficulty in applying control
theoretic methods to practical epidemiology problems is the
commonly made assumption that one has total knowledge of the state
of the epidemics \cite{dietz}.
\section{Statement of the problem}
\label{statement}
We denote by $S(t)$, $I(t)$, $R(t)$ the number of susceptible,
infectives and removed in a closed population of size $N$ at time
$t$. We assume the controlled dynamics
\begin{equation} \label{model}
\begin{gathered}
\dot S=- u_1 S I \, ,\\
\dot I=u_1 S I - u_2 I \,,\\
\dot R =u_2 I,
\end{gathered}
\end{equation}
The above models assume a mass-action type interaction ( for more
realistic interactions see \cite{chavez}). We let positive
constants $\beta$ and $\gamma$ denote the infection and removal
rates respectively without the influence of an education campaign.
Our controls are
$u_1(t) , u_2(t) $ with $u_1(t) \in [ \beta_m , \beta] $ and
$u_2(t) \in [ \gamma , \gamma_M]$ with
$0